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Chapter 7

Chapter 7. 7.1 Midpoint of the Line Joining Two Points. Coordinate Geometry. 7.2 Areas of Triangles and Quadrilaterals. 7.3 Parallel and Non-Parallel Lines. 7.4 Perpendicular Lines. Coordinate Geometry. 7.1 Midpoint of the Line Joining Two Points. Objectives.

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Chapter 7

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  1. Chapter 7 7.1 Midpoint of the Line Joining Two Points Coordinate Geometry 7.2 Areas of Triangles and Quadrilaterals 7.3 Parallel and Non-Parallel Lines 7.4 Perpendicular Lines

  2. Coordinate Geometry 7.1 Midpoint of the Line Joining Two Points Objectives In this lesson, you will learn how to find the midpoint of a line segment and apply it to solve problems.

  3. D is and E is M is the point Coordinate Geometry A line AB joins points (x1, y1) and (x2, y2). M (x, y) is the midpoint of AB. Take the x-coordinate of D and the y-coordinate of E. Construct a right angled triangle ABC. Construct the midpoints D and E of the line segments AC and BC. Take the mean of the coordinates at the endpoints.

  4. Coordinate Geometry Example 4 P, Q, R and S are coordinates of a parallelogram and M is the midpoint of PR. Find the coordinates of M and S and show that PQRS is a rhombus. M is also the midpoint of QS.

  5. Coordinate Geometry Exercise 7.1, qn 3 3 points have coordinates A(–1, 6), B(3, 2) and C(–5, –4). Given that D and E are the midpoints of AB and AC respectively, calculate the midpoint and length of DE. Let M be the midpoint of DE.

  6. Coordinate Geometry Exercise 7.1, qn 4 If A(2, 0), B(p, –2), C(–1, 1) and D(3, r) are the vertices of a parallelogram ABCD, calculate the values of p and r. Let M be the midpoint of AC. M is also the midpoint of BD.

  7. Coordinate Geometry 7.2 Areas of Triangles and Quadrilaterals Objectives In this lesson, you will learn how to find the areas of rectilinear figures given their vertices.

  8. Coordinate Geometry Area of Triangles ABC is a triangle. We will find its area. Construct points D and E so that ADEC is a trapezium.

  9. Coordinate Geometry ABC is a triangle. The vertices are arranged in an anticlockwise direction. We will find its area. Construct points D, E and F on the x-axis as shown.

  10. Coordinate Geometry Definition From the previous slide, we know that

  11. Coordinate Geometry Example 5 Find the area of a triangle with vertices A(–2, –1), B(2, –3) and C(4, 3). The vertices A, B and C follow an anticlockwise direction.

  12. Coordinate Geometry Area of Quadrilaterals Find the area of a quadrilateral with vertices A(x1, y1 ), B(x2, y2 ), C(x3, y3 ) and D(x4, y4 ), following an anticlockwise direction. Split the quadrilateral into two triangles.

  13. Coordinate Geometry The area of a quadrilateral with vertices A(x1, y1 ), B(x2, y2 ), C(x3, y3 ) and D(x4, y4 ), following an anticlockwise direction. The method for finding the area of quadrilaterals is very similar to that of triangles.

  14. Coordinate Geometry Exercise 7.2, qn 2(b) Find the area of a quadrilateral with vertices P(1, 4), Q(–4, 3), R(1, –2) and S(4, 0), following an anticlockwise direction.

  15. Coordinate Geometry 7.3 Parallel and Non-Parallel Lines Objectives In this lesson, you will learn how to apply the conditions for the gradients of parallel lines to solve problems.

  16. Coordinate Geometry Consider the straight line with equation y = m1 x + c1 that makes an angle of θ1 with the positive x-axis. Translate the line parallel to the x-axis. The new line has equation y = m2 x + c2 and makes an angle of θ2 with the positive x-axis. The lines are parallel to each other. θ1 = θ2 The lines make the same angle with the x-axis. m1 = m2 The lines have the same gradient.

  17. Coordinate Geometry Example 7(a) The diagram shows a parallelogram ABCD with A and C on the x-axis and y-axis respectively. The equation of AB is x + y = 2 and the equation of BC is 2y = x + 10. (a) Find the coordinates of A, B and C.

  18. Coordinate Geometry Example 7(b) The diagram shows a parallelogram ABCD with A and C on the x-axis and y-axis respectively. The equation of AB is x + y = 2 and the equation of BC is 2y = x + 10. (b) Find the equations of AD and CD. AD is parallel to BC (2y = x + 10). CD is parallel to AB (x + y = 2 ). Gradient of AD = gradient of BC = 0.5 Gradient of CD = gradient of AB = –1 Since A is (2, 0) Since C is (0, 5)

  19. Coordinate Geometry Exercise 7.3, qn 2(b) Find the equation of the line which passes through the point (–2, 3) and is parallel to the line 2x + 3y – 3 = 0. m in y = mx + c is the gradient of the line. Rearrange in the form y = mx + c.

  20. Coordinate Geometry 7.4 Perpendicular Lines Objectives In this lesson, you will learn how to apply the conditions for the gradients of perpendicular lines to solve problems.

  21. Coordinate Geometry Consider the straight line with equation y = m1 x + c1 that makes an angle of θ1 with the positive x-axis. Rotate the line clockwise through 90°. The new line has equation y = m2 x + c2 and makes an angle of θ2 with the negative x-axis. Applies to any two perpendicular lines.

  22. Coordinate Geometry Example 14 Two points have coordinates A(–2, 3) and B(4, 15). Find the equation of the perpendicular bisector of AB. Hence calculate the coordinates of the point P on the line 3y = x + 1 if P is equidistant from A and B. Solve simultaneous equations Find P. Adding the equations Substitute for y

  23. Coordinate Geometry Exercise 7.4, qn 9 The points A and B have coordinates (5, 2) and (3, 6) respectively. P and Q are points on the x-axis and y-axis and both P and Q are equidistant from A and B. • Find the equation of the perpendicular bisector of AB. (b) Find the coordinates of P and Q.

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